Fluid analysis of Markov Models

Transcription

1 Fluid analysis of Markov Models Jeremy Bradley, Richard Hayden, Anton Stefanek Imperial College London Tutorial, SFM:DS June 2013

2 2/60 You can download this presentation now from: or

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4 How can we... 4/60

5 scale 4/60

6 resource 4/60

7 provision 4/60

8 design 4/60

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10 or 5/60

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12 to meet 5/60

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14 while minimising 5/60

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16 ? 5/60

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18 We want to be able to engineer complex systems 6/60

19 6/60 We want to be able to engineer complex systems We want to be able to reason about performance

20 6/60 We want to be able to engineer complex systems We want to be able to reason about performance We want to be able to optimise key cost functions

21 6/60 We want to be able to engineer complex systems We want to be able to reason about performance We want to be able to optimise key cost functions...at the same time

22 Process modelling with Stochastic systems 7/60

23 8/60 Process Algebra A process algebra model consists of agents which engage in actions.

24 8/60 Process Algebra A process algebra model consists of agents which engage in actions. action label α.p agent or component Based on a slide by Jane Hillston

25 8/60 Process Algebra A process algebra model consists of agents which engage in actions. action label α.p agent or component Based on a slide by Jane Hillston

26 8/60 Process Algebra A process algebra model consists of agents which engage in actions. action label α.p agent or component The semantics of the language define an underlying state space by way of a labelled transition system. Based on a slide by Jane Hillston

27 8/60 Process Algebra A process algebra model consists of agents which engage in actions. action label α.p agent or component The semantics of the language define an underlying state space by way of a labelled transition system. A process algebra model Based on a slide by Jane Hillston

28 8/60 Process Algebra A process algebra model consists of agents which engage in actions. action label α.p agent or component The semantics of the language define an underlying state space by way of a labelled transition system. A process algebra model semantic rules Based on a slide by Jane Hillston

29 8/60 Process Algebra A process algebra model consists of agents which engage in actions. action label α.p agent or component The semantics of the language define an underlying state space by way of a labelled transition system. A process algebra model semantic rules Labelled transition system Based on a slide by Jane Hillston

30 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. 9/60

31 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. (α, λ).p action label agent or component action rate Based on a slide by Jane Hillston 9/60

32 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. (α, λ).p action label agent or component action rate Based on a slide by Jane Hillston 9/60

33 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. (α, λ).p action label agent or component action rate Based on a slide by Jane Hillston 9/60

34 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. (α, λ).p action label agent or component action rate where the action duration is exponentially distributed with rate λ. Based on a slide by Jane Hillston 9/60

35 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. (α, λ).p action label agent or component action rate where the action duration is exponentially distributed with rate λ. The semantics of the language define an underlying state space and also a performance model in terms of a CTMC. Based on a slide by Jane Hillston 9/60

36 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. (α, λ).p action label agent or component action rate where the action duration is exponentially distributed with rate λ. The semantics of the language define an underlying state space and also a performance model in terms of a CTMC. SPA model Based on a slide by Jane Hillston 9/60

37 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. (α, λ).p action label agent or component action rate where the action duration is exponentially distributed with rate λ. The semantics of the language define an underlying state space and also a performance model in terms of a CTMC. SPA model semantics Based on a slide by Jane Hillston 9/60

38 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. (α, λ).p action label agent or component action rate where the action duration is exponentially distributed with rate λ. The semantics of the language define an underlying state space and also a performance model in terms of a CTMC. SPA model semantics LTS Based on a slide by Jane Hillston 9/60

39 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. (α, λ).p action label agent or component action rate where the action duration is exponentially distributed with rate λ. The semantics of the language define an underlying state space and also a performance model in terms of a CTMC. SPA model semantics LTS filter Based on a slide by Jane Hillston 9/60

40 Stochastic Process Algebra A stochastic process algebra model also consists of agents which engage in actions, but with the actions having a random duration associated with them. (α, λ).p action label agent or component action rate where the action duration is exponentially distributed with rate λ. The semantics of the language define an underlying state space and also a performance model in terms of a CTMC. semantics filter SPA model LTS CTMC Based on a slide by Jane Hillston 9/60

41 PEPA: Stochastic process algebra Many SPAs exist and capture performance and behavioural features in different ways. e.g. igsmpa [1], IMC [2], sfsp [3], EMPA [4], TIPP [5] PEPA [6] is useful because: it is a formal, algebraic description of a system it is compositional it is parsimonious (succinct) it is easy to learn! it is used in research and in industry [1] Mario Bravetti and Roberto Gorrieri. Interactive Generalized Semi-Markov Processes. In: Process Algebra and Performance Modelling Workshop. Ed. by Jane Hillston and Manuel Silva. Centro Politécnico Superior de la Universidad de Zaragoza. Prensas Universitarias de Zaragoza, 1999, pp [2] Holger Hermanns. Interactive Markov Chains. PhD thesis. Universität Erlangen Nürnberg, [3] Thomas Ayles et al. Adding Performance Evaluation to the LTSA Tool. In: Proceedings of 13th International Conference on Computer Performance Evaluation: Modelling Techniques and Tools [4] Marco Bernardo and Roberto Gorrieri. Extended Markovian Process Algebra. In: CONCUR 96, Proceedings of the 7th International Conference on Concurrency Theory. Ed. by Ugo Montanari and Vladimiro Sassone. Vol Lecture Notes in Computer Science. Springer- Verlag, 1996, pp [5] Norbert Götz, Ulrich Herzog, and Michael Rettelbach. TIPP A Stochastic Process Algebra. In: Process Algebra and Performance Modelling. Ed. by Jane Hillston and Faron Moller. CSR Technical Report. Department of Computer Science, University of Edinburgh, 1993, pp [6] Jane Hillston. A Compositional Approach to Performance Modelling. Vol. 12. Distinguished Dissertations in Computer Science. Cambridge University Press, /60

42 What can you do with PEPA? It allows you to answer key performance questions Steady state analysis 0.05 Steady state: X_ Probability Time, t What is the long-run average behaviour of my system? 11/60

43 What can you do with PEPA? It allows you to answer key performance questions Transient analysis PEPA model: transient X_1 -> X_1 Steady state: X_ Probability Time, t What is the behaviour of my system at time, t? 11/60

44 What can you do with PEPA? It allows you to answer key performance questions Transient analysis PEPA model: transient X_1 -> X_1 Steady state: X_ Probability Time, t What is the behaviour of my system at time, t? 11/60

45 What can you do with PEPA? It allows you to answer key performance questions Transient analysis PEPA model: transient X_1 -> X_1 Steady state: X_ Probability Time, t What is the behaviour of my system at time, t? 11/60

46 11/60 What can you do with PEPA? It allows you to answer key performance questions Passage time analysis How long does it take my system to complete a key transaction?

47 Tool Support PEPA has several methods of execution and analysis, through comprehensive tool support: PEPA Eclipse plugin: Edinburgh [7] Möbius: Urbana-Champaign, Illinois [8] PRISM: Birmingham [9] ipc: Imperial College London [10] gpa: Imperial College London [11] [7] Mirco Tribastone. The PEPA Plug-in Project. In: QEST 07, Proceedings of the 4th Int. Conference on the Quantitative Evaluation of Systems. IEEE Computer Society, 2007, pp [8] Graham Clark et al. The Möbius Modeling Tool. In: Proceedings the 9th International Workshop on Petri Nets and Performance Models. Ed. by B Haverkort and R German. IEEE Computer Society Press, 2001, pp [9] Marta Kwiatkowska, Gethin Norman, and David Parker. PRISM: Probabilistic Symbolic Model Checker. In: TOOLS 02, Proceedings of the 12th International Conference on Modelling Techniques and Tools for Computer Performance Evaluation. Ed. by A J Field et al. Vol Lecture Notes in Computer Science. London: Springer-Verlag, 2002, pp [10] Jeremy T Bradley and William J Knottenbelt. The ipc/hydra Tool Chain for the Analysis of PEPA Models. In: QEST 04, Proceedings of the 1st IEEE Conference on the Quantitative Evaluation of Systems. Ed. by Boudewijn Haverkort et al. University of Twente, Enschede: IEEE Computer Society, 2004, pp [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. A new tool for the performance analysis of massively parallel computer systems. In: Eighth Workshop on Quantitative Aspects of Programming Languages (QAPL 2010), March 27-28, 2010, Paphos, Cyprus. Electronic Proceedings in Theoretical Computer Science. Mar url: 12/60

48 13/60 PEPA Syntax Syntax: def P ::= (a, λ).p P + P P P P/L A = P L Action prefix: (a, λ).p Competitive choice: P 1 + P 2 Cooperation: P 1 L P 2 Action hiding: P/L Constant label: A = def P

49 13/60 PEPA Syntax Syntax: def P ::= (a, λ).p P + P P P P/L A = P L Action prefix: (a, λ).p Competitive choice: P 1 + P 2 Cooperation: P 1 L P 2 Action hiding: P/L Constant label: A = def P

50 13/60 PEPA Syntax Syntax: def P ::= (a, λ).p P + P P P P/L A = P L Action prefix: (a, λ).p Competitive choice: P 1 + P 2 Cooperation: P 1 L P 2 Action hiding: P/L Constant label: A = def P

51 13/60 PEPA Syntax Syntax: def P ::= (a, λ).p P + P P P P/L A = P L Action prefix: (a, λ).p Competitive choice: P 1 + P 2 Cooperation: P 1 L P 2 Action hiding: P/L Constant label: A = def P

52 13/60 PEPA Syntax Syntax: def P ::= (a, λ).p P + P P P P/L A = P L Action prefix: (a, λ).p Competitive choice: P 1 + P 2 Cooperation: P 1 L P 2 Action hiding: P/L Constant label: A = def P

53 13/60 PEPA Syntax Syntax: def P ::= (a, λ).p P + P P P P/L A = P L Action prefix: (a, λ).p Competitive choice: P 1 + P 2 Cooperation: P 1 L P 2 Action hiding: P/L Constant label: A = def P

54 14/60 Why exponential? 1.4 X~exp(1.25)

55 14/60 Why exponential? Memorylessness 1.4 X~exp(1.25) What is P(X t X > u) if X exp(λ)?

56 14/60 Why exponential? Memorylessness 1.4 X~exp(1.25) What is P(X t X > u) if X exp(λ)?

57 14/60 Why exponential? Memorylessness 1.4 X~exp(1.25) What is P(X t X > u) if X exp(λ)?

58 14/60 Why exponential? Memorylessness 1.4 X~exp(1.25) What is P(X t X > u) if X exp(λ)?

59 14/60 Why exponential? Memorylessness 1.4 X~exp(1.25) What is P(X t X > u) if X exp(λ)?

60 14/60 Why exponential? Memorylessness 1.4 X~exp(1.25) What is P(X t X > u) if X exp(λ)?

61 14/60 Why exponential? Memorylessness 1.4 X~exp(1.25) What is P(X t X > u) if X exp(λ)?

62 14/60 Why exponential? Memorylessness 1.4 X~exp(1.25) What is P(X t X > u) if X exp(λ)?

63 15/60 Why exponential? 1. Described by a single parameter

64 15/60 Why exponential? 1. Described by a single parameter 2. Memorylessness

65 15/60 Why exponential? 1. Described by a single parameter 2. Memorylessness 3. Ability to describe other distributions using phase-type combinations

66 16/60 Prefix: (a, λ).a Prefix is used to describe a process that evolves from one state to another by emitting or performing an action

67 16/60 Prefix: (a, λ).a Prefix is used to describe a process that evolves from one state to another by emitting or performing an action Example: P = def (a, λ).q...means that the process P evolves with rate λ to become process Q, by emitting an a-action

68 16/60 Prefix: (a, λ).a Prefix is used to describe a process that evolves from one state to another by emitting or performing an action Example: P = def (a, λ).q...means that the process P evolves with rate λ to become process Q, by emitting an a-action λ is an exponential rate parameter

69 16/60 Prefix: (a, λ).a Prefix is used to describe a process that evolves from one state to another by emitting or performing an action Example: P = def (a, λ).q...means that the process P evolves with rate λ to become process Q, by emitting an a-action λ is an exponential rate parameter As a labelled transition system, this becomes: Prefix : P (a,λ) Q

70 17/60 Choice: P 1 + P 2 PEPA uses a type of choice known as competitive choice

71 17/60 Choice: P 1 + P 2 PEPA uses a type of choice known as competitive choice Example: P = def (a, λ).p 1 + (b, µ).p 2 means that P can evolve either to produce an a-action with rate λ or to produce a b-action with rate µ

72 17/60 Choice: P 1 + P 2 PEPA uses a type of choice known as competitive choice Example: P = def (a, λ).p 1 + (b, µ).p 2 means that P can evolve either to produce an a-action with rate λ or to produce a b-action with rate µ As a labelled transition system: (a, λ) Choice: P (b, µ) P 1 P 2

73 18/60 Choice: P 1 + P 2 P = def (a, λ).p 1 + (b, µ).p 2 This is competitive choice since: P 1 and P 2 are in a race condition the first one to perform an a or a b will dictate the direction of choice for P 1 + P 2 What is the probability that we see an a-action?

74 19/60 Cooperation: P 1 L P 2 P 1 P 2 defines concurrency and communication within L PEPA

75 19/60 Cooperation: P 1 L P 2 P 1 P 2 defines concurrency and communication within L PEPA The L in P 1 P 2 defines the set of actions over which two L components are to cooperate

76 19/60 Cooperation: P 1 L P 2 P 1 P 2 defines concurrency and communication within L PEPA The L in P 1 P 2 defines the set of actions over which two L components are to cooperate Any other actions that P 1 and P 2 can do, not mentioned in L, can happen independently

77 19/60 Cooperation: P 1 L P 2 P 1 P 2 defines concurrency and communication within L PEPA The L in P 1 P 2 defines the set of actions over which two L components are to cooperate Any other actions that P 1 and P 2 can do, not mentioned in L, can happen independently If a L and P 1 enables an a, then P 1 has to wait for P 2 to enable an a before the cooperation can proceed

78 19/60 Cooperation: P 1 L P 2 P 1 P 2 defines concurrency and communication within L PEPA The L in P 1 P 2 defines the set of actions over which two L components are to cooperate Any other actions that P 1 and P 2 can do, not mentioned in L, can happen independently If a L and P 1 enables an a, then P 1 has to wait for P 2 to enable an a before the cooperation can proceed Easy source of deadlock!

79 20/60 Cooperation: P 1 L P 2 If P 1 (a,λ) P 1 and P 2 (a, ) P 2 then: P 1 {a} P 2 (a,λ) P 1 P {a} 2

80 20/60 Cooperation: P 1 L P 2 If P 1 (a,λ) P 1 and P 2 (a, ) P 2 then: P 1 {a} P 2 (a,λ) P 1 P {a} 2 represents a passive rate which, in the cooperation, inherits the λ-rate of from P 1

81 20/60 Cooperation: P 1 L P 2 If P 1 (a,λ) P 1 and P 2 (a, ) P 2 then: P 1 {a} P 2 (a,λ) P 1 P {a} 2 represents a passive rate which, in the cooperation, inherits the λ-rate of from P 1 If both rates are specified and the only a-evolutions allowed from P 1 and P 2 are, P 1 P 1 P (a,min(λ,µ)) 2 P {a} 1 P {a} 2 (a,λ) P 1 and P 2 (a,µ) P 2 then:

82 21/60 Cooperation: P 1 L P 2 The general cooperation case is where: P1 enables m a-actions P 2 enables n a-actions at the moment of cooperation...in which case there are m n possible transitions for P 1 {a} P 2

83 21/60 Cooperation: P 1 L P 2 The general cooperation case is where: P1 enables m a-actions P 2 enables n a-actions at the moment of cooperation...in which case there are m n possible transitions for P 1 {a} P 2 with mn a-actions having cumulative rate P 1 P 2 {a} where R = min(r a (P 1 ), r a (P 2 )) (a,r)

84 21/60 Cooperation: P 1 L P 2 The general cooperation case is where: P1 enables m a-actions P 2 enables n a-actions at the moment of cooperation...in which case there are m n possible transitions for P 1 {a} P 2 with mn a-actions having cumulative rate P 1 P 2 {a} where R = min(r a (P 1 ), r a (P 2 )) (a,r) r a (P) = i:p (a, r i ) r i is the apparent rate of an action a the total rate at which P can do a

85 Hiding: P/L Used to turn observable actions in P into hidden or silent actions in P/L L defines the set of actions to hide If P (a,λ) P : P/{a} (τ,λ) P /{a} τ is the silent action Used to hide complexity and create a component interface Cooperation on τ not allowed 22/60

86 23/60 PEPA: A Transmitter-Receiver System = def (Transmitter Receiver) Network L A simple model of a transmitter receiver over a network

87 23/60 PEPA: A Transmitter-Receiver System = def (Transmitter Receiver) Network L A simple model of a transmitter receiver over a network

88 23/60 PEPA: A Transmitter-Receiver System = def (Transmitter Receiver) Network Transmitter = def (transmit, λ 1 ).(t recover, λ 2 ).Transmitter Receiver = def (receive, ).(r recover, µ).receiver Network = def (transmit, ).(delay, ν 1 ).(receive, ν 2 ).Network L A simple model of a transmitter receiver over a network

89 23/60 PEPA: A Transmitter-Receiver System = def (Transmitter Receiver) Network Transmitter = def (transmit, λ 1 ).(t recover, λ 2 ).Transmitter Receiver = def (receive, ).(r recover, µ).receiver Network = def (transmit, ).(delay, ν 1 ).(receive, ν 2 ).Network where L = {transmit, receive}. L A simple model of a transmitter receiver over a network

90 TR example: Labelled transition system with X 1 (Transmitter Receiver) Network L X 2 (Transmitter Receiver) Network and so on. L 24/60

91 25/60 Voting Example I Voters vote and Pollers record those votes. Pollers can break individually and recover individually. If all Pollers break then they are all repaired in unison. System def = (Voter Voter Voter) ((Poller Poller) Poller group 0) {vote} L L where L = {recover all} L = {recover, break, recover all}

92 26/60 Voting Example II Voter def = (vote, λ).(pause, µ).voter

93 26/60 Voting Example II Voter def = (vote, λ).(pause, µ).voter Poller def = (vote, ).(register, γ).poller + (break, ν).poller broken

94 26/60 Voting Example II Voter def = (vote, λ).(pause, µ).voter Poller def = (vote, ).(register, γ).poller + (break, ν).poller broken Poller broken def = (recover, τ).poller + (recover all, ).Poller

95 27/60 Voting Example III Poller group 0 def = (break, ).Poller group 1

96 27/60 Voting Example III Poller group 0 def = (break, ).Poller group 1 Poller group 1 def = (break, ).Poller group 2 + (recover, ).Poller group 0

97 27/60 Voting Example III Poller group 0 def = (break, ).Poller group 1 Poller group 1 def = (break, ).Poller group 2 + (recover, ).Poller group 0 Poller group 2 def = (recover all, δ).poller group 0

98 An Overview of model-based Fluid Analysis 28/60

99 Mean field/fluid analysis Addresses the state-space explosion problem for discrete-state Markov models of computer and communication systems [12] Jane Hillston. Fluid flow approximation of PEPA models. In: Second International Conference on the Quantitative Evaluation of Systems (QEST). IEEE, Sept. 2005, pp doi: /QEST [13] Michel Benaïm and Jean-Yves Le Boudec. A class of mean field interaction models for computer and communication systems. In: Performance Evaluation (Nov. 2008), pp doi: /j.peva [14] Marco Gribaudo. Analysis of Large Populations of Interacting Objects with Mean Field and Markovian Agents. In: 6th European Performance Engineering Workshop (EPEW). Vol , pp doi: / /60

100 Mean field/fluid analysis Addresses the state-space explosion problem for discrete-state Markov models of computer and communication systems Derives tractable systems of differential equations approximating mean number of components in each local state, for example: Fluid analysis of process algebra models [12] Mean-field analysis of systems of interacting objects [13,14] [12] Jane Hillston. Fluid flow approximation of PEPA models. In: Second International Conference on the Quantitative Evaluation of Systems (QEST). IEEE, Sept. 2005, pp doi: /QEST [13] Michel Benaïm and Jean-Yves Le Boudec. A class of mean field interaction models for computer and communication systems. In: Performance Evaluation (Nov. 2008), pp doi: /j.peva [14] Marco Gribaudo. Analysis of Large Populations of Interacting Objects with Mean Field and Markovian Agents. In: 6th European Performance Engineering Workshop (EPEW). Vol , pp doi: / /60

101 Mean field/fluid analysis Addresses the state-space explosion problem for discrete-state Markov models of computer and communication systems Derives tractable systems of differential equations approximating mean number of components in each local state, for example: Fluid analysis of process algebra models [12] Mean-field analysis of systems of interacting objects [13,14] Can develop these techniques to capture key performance measures of interest from large CTMCs, e.g. passage-time measures, reward-based measures [12] Jane Hillston. Fluid flow approximation of PEPA models. In: Second International Conference on the Quantitative Evaluation of Systems (QEST). IEEE, Sept. 2005, pp doi: /QEST [13] Michel Benaïm and Jean-Yves Le Boudec. A class of mean field interaction models for computer and communication systems. In: Performance Evaluation (Nov. 2008), pp doi: /j.peva [14] Marco Gribaudo. Analysis of Large Populations of Interacting Objects with Mean Field and Markovian Agents. In: 6th European Performance Engineering Workshop (EPEW). Vol , pp doi: / /60

102 A simple agent 30/60

103 A simple agent replicated 30/60

104 A simple agent replicated 30/60

105 A simple agent replicated 30/60

106 A simple agent replicated 30/60

107 A simple agent replicated 30/60

108 A simple agent replicated 30/60

109 A simple agent replicated 30/60

110 A simple agent replicated 30/60

111 A simple agent replicated 30/60

112 A simple agent replicated 30/60

113 30/60 A simple agent replicated Fluid/mean field analysis works best when you have many replicated parallel agents or groups of replicated parallel agents. Agent groups can synchronise.

114 31/60 GPEPA Syntax GPEPA or Grouped PEPA as a syntax that is suspiciously similar to that of PEPA.

115 31/60 GPEPA Syntax GPEPA or Grouped PEPA as a syntax that is suspiciously similar to that of PEPA. An sequential agent, P, can have the following syntax:

116 31/60 GPEPA Syntax GPEPA or Grouped PEPA as a syntax that is suspiciously similar to that of PEPA. An sequential agent, P, can have the following syntax: P ::= (a, λ).p SPA Markovian prefix

117 31/60 GPEPA Syntax GPEPA or Grouped PEPA as a syntax that is suspiciously similar to that of PEPA. An sequential agent, P, can have the following syntax: P ::= (a, λ).p SPA Markovian prefix P + P SPA competitive choice

118 31/60 GPEPA Syntax GPEPA or Grouped PEPA as a syntax that is suspiciously similar to that of PEPA. An sequential agent, P, can have the following syntax: P ::= (a, λ).p SPA Markovian prefix P + P SPA competitive choice C = def P Agent name

119 31/60 GPEPA Syntax GPEPA or Grouped PEPA as a syntax that is suspiciously similar to that of PEPA. An sequential agent, P, can have the following syntax: P ::= (a, λ).p SPA Markovian prefix P + P SPA competitive choice C = def P Agent name Sequential agents allow a modeller to define behaviour with associated exponential delays.

120 32/60 GPEPA Syntax For parallelism and communication between sequential agents, we need compositional agents.

121 32/60 GPEPA Syntax For parallelism and communication between sequential agents, we need compositional agents. A compositional agent, Q, can have the following syntax:

122 32/60 GPEPA Syntax For parallelism and communication between sequential agents, we need compositional agents. A compositional agent, Q, can have the following syntax: Q ::= Q Q L Group cooperation Group{P[n]} Parallel grouping where P[n] represents a parallel group of n sequential agents P. Group represents a group label used to identify the parts of the model that are going to be approximated using fluid analysis.

123 33/60 GPEPA Example Client = def (req, r req ).Client waiting Client waiting = def (data, r data ).Client think Client think = def (think, r think ).Client

124 33/60 GPEPA Example Client = def (req, r req ).Client waiting Client waiting = def (data, r data ).Client think Client think = def (think, r think ).Client Server = def (req, r req ).Server get + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server

125 33/60 GPEPA Example Client = def (req, r req ).Client waiting Client waiting = def (data, r data ).Client think Client think = def (think, r think ).Client Server = def (req, r req ).Server get + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

126 33/60 GPEPA Example Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think Client think = def (think, r think ).Client Server = def (req, r req ).Server get + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

127 33/60 GPEPA Example Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think C w (t) Client think = def (think, r think ).Client Server = def (req, r req ).Server get + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

128 33/60 GPEPA Example Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think C w (t) Client think = def (think, r think ).Client C t (t) Server = def (req, r req ).Server get + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

129 33/60 GPEPA Example Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think C w (t) Client think = def (think, r think ).Client C t (t) Server = def (req, r req ).Server get S(t) + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

130 33/60 GPEPA Example Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think C w (t) Client think = def (think, r think ).Client C t (t) Server = def (req, r req ).Server get S(t) + (break, r break ).Server broken Server get = def (data, r data ).Server S g (t) Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

131 33/60 GPEPA Example Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think C w (t) Client think = def (think, r think ).Client C t (t) Server = def (req, r req ).Server get S(t) + (break, r break ).Server broken Server get = def (data, r data ).Server S g (t) Server broken = def (reset, r reset ).Server S b (t) CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

132 34/60 ODEs Means Ideally, we want the distribution of say C(t) for each t

133 34/60 ODEs Means Ideally, we want the distribution of say C(t) for each t This can be too expensive

134 34/60 ODEs Means Ideally, we want the distribution of say C(t) for each t This can be too expensive Can derive ODEs approximating the means de[c(t)]/dt = de[s(t)]/dt = de[c w (t)]/dt = de[s g (t)]/dt = de[c t (t)]/dt = de[s b (t)]/dt =

135 34/60 ODEs Means Ideally, we want the distribution of say C(t) for each t This can be too expensive Can derive ODEs approximating the means de[c(t)]/dt = de[s(t)]/dt = de[c w (t)]/dt = de[s g (t)]/dt = de[c t (t)]/dt = de[s b (t)]/dt = These can be numerically solved, cheaper than simulation

136 35/60 ODEs Means ee[c(t)] 60 ee[s(t)] Count ee[c w (t)] ee[c t(t)] ee[s g (t)] ee[s b(t)] Time, t Time, t

137 36/60 ODEs Higher moments Can extend the ODEs de[c(t)]/dt = de[s(t)]/dt = de[c w (t)]/dt = de[s g (t)]/dt = de[c t (t)]/dt = de[s b (t)]/dt =

138 36/60 ODEs Higher moments Can extend the ODEs de[c(t)]/dt = de[s(t)]/dt = de[c w (t)]/dt = de[s g (t)]/dt = de[c t (t)]/dt = de[s b (t)]/dt = with ODEs for higher moments de[s g (t) 2 ]/dt = de[c(t)s(t)]/dt =.

139 36/60 ODEs Higher moments Can extend the ODEs de[c(t)]/dt = de[s(t)]/dt = de[c w (t)]/dt = de[s g (t)]/dt = de[c t (t)]/dt = de[s b (t)]/dt = with ODEs for higher moments de[s g (t) 2 ]/dt = de[c(t)s(t)]/dt = E.g. can get variance as. Var[C(t)] = E[C(t) 2 ] E[C(t)] 2

140 37/60 ODEs Higher moments ee[c(t)] 60 ee[s(t)] Count ee[c w (t)] ee[c t(t)] ee[s g (t)] ee[s b(t)] Time, t Time, t

141 37/60 ODEs Higher moments ee[c(t)] 60 ee[s(t)] Count ee[c w (t)] ee[c t(t)] ee[s g (t)] ee[s b(t)] Time, t Time, t

142 38/60 Accumulated rewards How to analyse quantities accumulated over time, e.g. energy consumption?

143 38/60 Accumulated rewards Server broken How to analyse quantities accumulated over time, e.g. energy consumption? Servers consume energy in the Server get state Server get state at t Server Time, t

144 38/60 Accumulated rewards Server broken How to analyse quantities accumulated over time, e.g. energy consumption? Servers consume energy in the Server get state Server get state at t Server Time, t

145 38/60 Accumulated rewards How to analyse quantities accumulated over time, e.g. energy consumption? Servers consume energy in the Server get state Server broken Server get Server state at t Count S g (t) Time, t Time, t

146 38/60 Accumulated rewards How to analyse quantities accumulated over time, e.g. energy consumption? Servers consume energy in the Server get state Server broken Server get Server state at t Count S g (t) Time, t Time, t

147 38/60 Accumulated rewards How to analyse quantities accumulated over time, e.g. energy consumption? Servers consume energy in the Server get state Server broken Server get Server state at t Count S g (t) Time, t The total energy consumption is the process t 0 S g (u)du Time, t

148 39/60 ODEs moments of rewards Can extend the ODEs for moments of counts de[c(t)]/dt = de[s(t) 2 ]/dt = de[c w (t)s g (t)]/dt = de[s g (t) 3 ]/dt =..

149 39/60 ODEs moments of rewards Can extend the ODEs for moments of counts de[c(t)]/dt = de[s(t) 2 ]/dt = de[c w (t)s g (t)]/dt = de[s g (t) 3 ]/dt = with ODEs for the mean accumulated rewards. [ ] t de 0 S g (u)du /dt = E[S g (t)] [ ] t de 0 S(u)C(u)du /dt = E[S(t)C(t)]..

150 39/60 ODEs moments of rewards Can extend the ODEs for moments of counts de[c(t)]/dt = de[s(t) 2 ]/dt = de[c w (t)s g (t)]/dt = de[s g (t) 3 ]/dt = with ODEs for the mean accumulated rewards. [ ] t de 0 S g (u)du /dt = E[S g (t)] [ ] t de 0 S(u)C(u)du /dt = E[S(t)C(t)]. and ODEs for higher moments of accumulated rewards [ ( ) ] t 2 de 0 S g (u)du /dt =..

151 40/60 ODEs moments of rewards 150 Mean reward ee[c(t)] Time, t

152 40/60 ODEs moments of rewards 150 Mean reward ee[c(t)] Time, t

153 GPA Grouped PEPA Analyser 41/60

154 42/60 Why tool? de[c(t)sg (t)]/dt+ = ( 1.0) (min((e[c(t)cw (t)]) (r data), (E[C(t)Sg (t)]) (r data))) de[c(t)cw (t)]/dt+ = ( 1.0) (min((e[c(t)cw (t)]) (r data), (E[C(t)Sg (t)]) (r data))) de[c(t)ct (t)]/dt+ = min((e[c(t)cw (t)]) (r data), (E[C(t)Sg (t)]) (r data)) de[sg (t)s(t)]/dt+ = min((e[sg (t)cw (t)]) (r data), (E[Sg (t) 2 ]) (r data)) de[sg (t) 2 ]/dt+ = ( 2.0) (min((e[sg (t)cw (t)]) (r data), (E[Sg (t) 2 ]) (r data))) de[sg (t)cw (t)]/dt+ = ( 1.0) (min((e[sg (t)cw (t)]) (r data), (E[Sg (t) 2 ]) (r data))) de[sg (t)ct (t)]/dt+ = min((e[sg (t)cw (t)]) (r data), (E[Sg (t) 2 ]) (r data)) de[s(t)ct (t)]/dt+ = min((e[cw (t)ct (t)]) (r data), (E[Sg (t)ct (t)]) (r data)) de[sg (t)ct (t)]/dt+ = ( 1.0) (min((e[cw (t)ct (t)]) (r data), (E[Sg (t)ct (t)]) (r data))) de[cw (t)ct (t)]/dt+ = ( 1.0) (min((e[cw (t)ct (t)]) (r data), (E[Sg (t)ct (t)]) (r data))) de[ct (t) 2 ]/dt+ = (2.0) (min((e[cw (t)ct (t)]) (r data), (E[Sg (t)ct (t)]) (r data))) de[cw (t)s(t)]/dt+ = min((e[cw (t) 2 ]) (r data), (E[Sg (t)cw (t)]) (r data)) de[sg (t)cw (t)]/dt+ = ( 1.0) (min((e[cw (t) 2 ]) (r data), (E[Sg (t)cw (t)]) (r data))) de[cw (t) 2 ]/dt+ = ( 2.0) (min((e[cw (t) 2 ]) (r data), (E[Sg (t)cw (t)]) (r data))) de[cw (t)ct (t)]/dt+ = min((e[cw (t) 2 ]) (r data), (E[Sg (t)cw (t)]) (r data)) de[s(t) 2 ]/dt+ = (2.0) ((E[S(t)S b(t)]) (rreset )) de[s(t)s b(t)]/dt+ = ( 1.0) ((E[S(t)S b(t)]) (rreset )) de[s(t)s b(t)]/dt+ = (E[S b(t) 2 ]) (rreset ) de[s b(t) 2 ]/dt+ = ( 2.0) ((E[S b(t) 2 ]) (rreset )) de[s(t)]/dt+ = (E[S b(t)]) (rreset ) de[s b(t)]/dt+ = ( 1.0) ((E[S b(t)]) (rreset )) de[s(t) 2 ]/dt+ = (E[S b(t)]) (rreset ) de[s(t)s b(t)]/dt+ = ( 1.0) ((E[S b(t)]) (rreset )) de[s b(t) 2 ]/dt+ = (E[S b(t)]) (rreset ) de[c(t)s(t)]/dt+ = (E[C(t)S b(t)]) (rreset ) de[c(t)s b(t)]/dt+ = ( 1.0) ((E[C(t)S b(t)]) (rreset )) de[sg (t)s(t)]/dt+ = (E[Sg (t)s b(t)]) (rreset ) de[sg (t)s b(t)]/dt+ = ( 1.0) ((E[Sg (t)s b(t)]) (rreset )) de[s(t)ct (t)]/dt+ = (E[Ct (t)s b(t)]) (rreset ) de[ct (t)s b(t)]/dt+ = ( 1.0) ((E[Ct (t)s b(t)]) (rreset )) de[cw (t)s(t)]/dt+ = (E[Cw (t)s b(t)]) (rreset ) de[cw (t)s b(t)]/dt+ = ( 1.0) ((E[Cw (t)s b(t)]) (rreset )) de[s(t)ct (t)]/dt+ = ( 1.0) (min((e[c(t)ct (t)]) (rreq), (E[S(t)Ct (t)]) (rreq))) de[sg (t)ct (t)]/dt+ = min((e[c(t)ct (t)]) (rreq), (E[S(t)Ct (t)]) (rreq)) de[c(t)ct (t)]/dt+ = ( 1.0) (min((e[c(t)ct (t)]) (rreq), (E[S(t)Ct (t)]) (rreq))) de[cw (t)ct (t)]/dt+ = min((e[c(t)ct (t)]) (rreq), (E[S(t)Ct (t)]) (rreq)) de[cw (t)s(t)]/dt+ = ( 1.0) (min((e[c(t)cw (t)]) (rreq), (E[Cw (t)s(t)]) (rreq))) de[sg (t)cw (t)]/dt+ = min((e[c(t)cw (t)]) (rreq), (E[Cw (t)s(t)]) (rreq)) de[c(t)cw (t)]/dt+ = ( 1.0) (min((e[c(t)cw (t)]) (rreq), (E[Cw (t)s(t)]) (rreq))) de[cw (t) 2 ]/dt+ = (2.0) (min((e[c(t)cw (t)]) (rreq), (E[Cw (t)s(t)]) (rreq))) de[s(t) 2 ]/dt+ = (2.0) (min((e[cw (t)s(t)]) (r data), (E[Sg (t)s(t)]) (r data))) de[sg (t)s(t)]/dt+ = ( 1.0) (min((e[cw (t)s(t)]) (r data), (E[Sg (t)s(t)]) (r data))) de[cw (t)s(t)]/dt+ = ( 1.0) (min((e[cw (t)s(t)]) (r data), (E[Sg (t)s(t)]) (r data))) de[s(t)ct (t)]/dt+ = min((e[cw (t)s(t)]) (r data), (E[Sg (t)s(t)]) (r data)) de[s(t)s b(t)]/dt+ = min((e[cw (t)s b(t)]) (r data), (E[Sg (t)s b(t)]) (r data)) de[sg (t)s b(t)]/dt+ = ( 1.0) (min((e[cw (t)s b(t)]) (r data), (E[Sg (t)s b(t)]) (r data))) de[cw (t)s b(t)]/dt+ = ( 1.0) (min((e[cw (t)s b(t)]) (r data), (E[Sg (t)s b(t)]) (r data))) de[ct (t)s b(t)]/dt+ = min((e[cw (t)s b(t)]) (r data), (E[Sg (t)s b(t)]) (r data)) de[s(t)]/dt+ = min((e[cw (t)]) (r data), (E[Sg (t)]) (r data)) de[sg (t)]/dt+ = ( 1.0) (min((e[cw (t)]) (r data), (E[Sg (t)]) (r data))) de[cw (t)]/dt+ = ( 1.0) (min((e[cw (t)]) (r data), (E[Sg (t)]) (r data))) de[ct (t)]/dt+ = min((e[cw (t)]) (r data), (E[Sg (t)]) (r data)) de[s(t) 2 ]/dt+ = min((e[cw (t)]) (r data), (E[Sg (t)]) (r data)) de[sg (t)s(t)]/dt+ = ( 1.0) (min((e[cw (t)]) (r data), (E[Sg (t)]) (r data))) de[cw (t)s(t)]/dt+ = ( 1.0) (min((e[cw (t)]) (r data), (E[Sg (t)]) (r data))) de[s(t)ct (t)]/dt+ = min((e[cw (t)]) (r data), (E[Sg (t)]) (r data)) de[sg (t) 2 ]/dt+ = min((e[cw (t)]) (r data), (E[Sg (t)]) (r data)) de[sg (t)cw (t)]/dt+ = min((e[cw (t)]) (r data), (E[Sg (t)]) (r data)) de[sg (t)ct (t)]/dt+ = ( 1.0) (min((e[cw (t)]) (r data), (E[Sg (t)]) (r data))) de[cw (t) 2 ]/dt+ = min((e[cw (t)]) (r data), (E[Sg (t)]) (r data)) de[cw (t)ct (t)]/dt+ = ( 1.0) (min((e[cw (t)]) (r data), (E[Sg (t)]) (r data))) de[ct (t) 2 ]/dt+ = min((e[cw (t)]) (r data), (E[Sg (t)]) (r data)) de[c(t)s(t)]/dt+ = min((e[c(t)cw (t)]) (r data), (E[C(t)Sg (t)]) (r data)) de[ct (t) 2 ]/dt+ = ( 2.0) ((E[Ct (t) 2 ]) (r think )) de[c(t)cw (t)]/dt+ = (E[Cw (t)ct (t)]) (r think ) de[cw (t)ct (t)]/dt+ = ( 1.0) ((E[Cw (t)ct (t)]) (r think )) de[s(t) 2 ]/dt+ = ( 2.0) (min((e[c(t)s(t)]) (rreq), (E[S(t) 2 ]) (rreq))) de[sg (t)s(t)]/dt+ = min((e[c(t)s(t)]) (rreq), (E[S(t) 2 ]) (rreq)) de[c(t)s(t)]/dt+ = ( 1.0) (min((e[c(t)s(t)]) (rreq), (E[S(t) 2 ]) (rreq))) de[cw (t)s(t)]/dt+ = min((e[c(t)s(t)]) (rreq), (E[S(t) 2 ]) (rreq)) de[s(t)s b(t)]/dt+ = ( 1.0) (min((e[c(t)s b(t)]) (rreq), (E[S(t)S b(t)]) (rreq))) de[sg (t)s b(t)]/dt+ = min((e[c(t)s b(t)]) (rreq), (E[S(t)S b(t)]) (rreq)) de[c(t)s b(t)]/dt+ = ( 1.0) (min((e[c(t)s b(t)]) (rreq), (E[S(t)S b(t)]) (rreq))) de[cw (t)s b(t)]/dt+ = min((e[c(t)s b(t)]) (rreq), (E[S(t)S b(t)]) (rreq)) de[s(t)]/dt+ = ( 1.0) (min((e[c(t)]) (rreq), (E[S(t)]) (rreq))) de[sg (t)]/dt+ = min((e[c(t)]) (rreq), (E[S(t)]) (rreq)) de[c(t)]/dt+ = ( 1.0) (min((e[c(t)]) (rreq), (E[S(t)]) (rreq))) de[cw (t)]/dt+ = min((e[c(t)]) (rreq), (E[S(t)]) (rreq)) de[s(t) 2 ]/dt+ = min((e[c(t)]) (rreq), (E[S(t)]) (rreq)) de[sg (t)s(t)]/dt+ = ( 1.0) (min((e[c(t)]) (rreq), (E[S(t)]) (rreq))) de[c(t)s(t)]/dt+ = min((e[c(t)]) (rreq), (E[S(t)]) (rreq)) de[cw (t)s(t)]/dt+ = ( 1.0) (min((e[c(t)]) (rreq), (E[S(t)]) (rreq))) de[sg (t) 2 ]/dt+ = min((e[c(t)]) (rreq), (E[S(t)]) (rreq)) de[c(t)sg (t)]/dt+ = ( 1.0) (min((e[c(t)]) (rreq), (E[S(t)]) (rreq))) de[sg (t)cw (t)]/dt+ = min((e[c(t)]) (rreq), (E[S(t)]) (rreq)) de[c(t) 2 ]/dt+ = min((e[c(t)]) (rreq), (E[S(t)]) (rreq)) de[c(t)cw (t)]/dt+ = ( 1.0) (min((e[c(t)]) (rreq), (E[S(t)]) (rreq))) de[cw (t) 2 ]/dt+ = min((e[c(t)]) (rreq), (E[S(t)]) (rreq)) de[c(t)s(t)]/dt+ = ( 1.0) (min((e[c(t) 2 ]) (rreq), (E[C(t)S(t)]) (rreq))) de[c(t)sg (t)]/dt+ = min((e[c(t) 2 ]) (rreq), (E[C(t)S(t)]) (rreq)) de[c(t) 2 ]/dt+ = ( 2.0) (min((e[c(t) 2 ]) (rreq), (E[C(t)S(t)]) (rreq))) de[c(t)cw (t)]/dt+ = min((e[c(t) 2 ]) (rreq), (E[C(t)S(t)]) (rreq)) de[sg (t)s(t)]/dt+ = ( 1.0) (min((e[sg (t)c(t)]) (rreq), (E[Sg (t)s(t)]) (rreq))) de[sg (t) 2 ]/dt+ = (2.0) (min((e[sg (t)c(t)]) (rreq), (E[Sg (t)s(t)]) (rreq))) de[c(t)sg (t)]/dt+ = ( 1.0) (min((e[sg (t)c(t)]) (rreq), (E[Sg (t)s(t)]) (rreq))) de[sg (t)cw (t)]/dt+ = min((e[sg (t)c(t)]) (rreq), (E[Sg (t)s(t)]) (rreq))

155 43/60 GPA GPEPA model

156 43/60 GPA GPEPA model generates ODEs

157 43/60 GPA GPEPA model generates ODEs num. solves Moments (approximate)

158 43/60 GPA GPEPA model generates ODEs num. solves Moments (approximate) Measures,passage times,etc.

159 43/60 GPA GPEPA model generates generates ODEs Simulation num. solves Moments (approximate) Measures,passage times,etc.

160 43/60 GPA GPEPA model generates generates ODEs num. solves Moments (approximate) Simulation Moments simulates Measures,passage times,etc.

161 43/60 GPA GPEPA model generates generates ODEs num. solves Moments (approximate) Simulation Moments simulates Measures,passage times,etc. Measures,passage times,etc.

162 43/60 GPA GPEPA model generates generates ODEs num. solves Moments (approximate) Error estimates Simulation Moments simulates Measures,passage times,etc. Measures,passage times,etc.

163 44/60 Grouped PEPA analyser Convenient syntax rreq = 2.0; rthink = 0.2;... c = 100.0; s = 50.0; Client = (request,rreq).client_waiting; Client_waiting = (data,rdata).client_think; Client_think = (think,rthink).client; Server Server_get Server_broken = (request,rreq).server_get + (break,rbreak).server_broken; = (data,rdata).server = (reset,rreset).server; Clients{Client[c]}<request,data>Servers{Server[s]}

164 45/60 GPA commands Analyses odes(stoptime=5.0, stepsize=0.01, density=10){...} simulation(stoptime=5.0,stepsize=0.01,repl.=1000){...} comparison(odes(...){...},simulation(...){...}){...}

165 45/60 GPA commands Analyses odes(stoptime=5.0, stepsize=0.01, density=10){...} simulation(stoptime=5.0,stepsize=0.01,repl.=1000){...} comparison(odes(...){...},simulation(...){...}){...} Plot commands, counts specified with Group:Component plot(e[clients:client],e[acc(clients:client)]); plot(e[acc(clients:client) Servers:Server_get^2]); plot(var[clients:client]);

166 45/60 GPA commands Analyses odes(stoptime=5.0, stepsize=0.01, density=10){...} simulation(stoptime=5.0,stepsize=0.01,repl.=1000){...} comparison(odes(...){...},simulation(...){...}){...} Plot commands, counts specified with Group:Component plot(e[clients:client],e[acc(clients:client)]); plot(e[acc(clients:client) Servers:Server_get^2]); plot(var[clients:client]); plot(e[clients:client]^2.0 + Var[Servers:Server]/s);

167 45/60 GPA commands Analyses odes(stoptime=5.0, stepsize=0.01, density=10){...} simulation(stoptime=5.0,stepsize=0.01,repl.=1000){...} comparison(odes(...){...},simulation(...){...}){...} Plot commands, counts specified with Group:Component plot(e[clients:client],e[acc(clients:client)]); plot(e[acc(clients:client) Servers:Server_get^2]); plot(var[clients:client]); plot(e[clients:client]^2.0 + Var[Servers:Server]/s); plotswitchpoints(1);

168 46/60 GPA passage times Allows general PEPA components NotPassed = (think,rthink).passed; Passed = (think,rthink).passed; ObservedClient = Client<think>NotPassed;

169 46/60 GPA passage times Allows general PEPA components NotPassed = (think,rthink).passed; Passed = (think,rthink).passed; ObservedClient = Client<think>NotPassed; For the CDF of first passage of a client E[C t P(t) + C w t P(t) + C t t P(t)]/c

170 46/60 GPA passage times Allows general PEPA components NotPassed = (think,rthink).passed; Passed = (think,rthink).passed; ObservedClient = Client<think>NotPassed; For the CDF of first passage of a client E[C t P(t) + C w t P(t) + C t t P(t)]/c Can use command plot(e[clients:_<*>passed]/c); For an upper bound on the CDF of first passage of 1/10-th of clients plot(var[clients:_<*>passed] /(Var[Clients:_<*>Passed]+(E[Clients:_<*>Passed]-c/10.0)^2.0));

171 47/60 GPA passage times 1 CDF 0.8 Probability Time t (a) Individual passage time for a client first passage Time t upper bound exact CDF lower bound (b) Global passage time until c/10 first passages

172 48/60 GPA completion times bounds(acc(servers:server get),100.0,2); 1 2 moments 0.8 Probability Time t completion time of t 0 S g (u)du reaching 100

173 48/60 GPA completion times bounds(acc(servers:server get),100.0,2,4); 1 2 moments 0.8 Probability Time t completion time of t 0 S g (u)du reaching 100

174 48/60 GPA completion times bounds(acc(servers:server get),100.0,2,4,6); 1 2 moments 0.8 Probability Time t completion time of t 0 S g (u)du reaching 100

175 49/60 Rapid analysis of very large scale parallel systems ODEs efficient implementation moments of: counts rewards passage and completion times error estimates large numbers of identical components described in GPEPA split-free splitting models Summary

176 49/60 Rapid analysis of very large scale parallel systems ODEs efficient implementation moments of: counts rewards passage and completion times error estimates large numbers of identical components described in GPEPA split-free splitting models Summary parallel solvers hybrid solutions impulse rewards time correlated moments optimisation nature of error new formalism complex synch. guards functional rates phase type Current work

177 49/60 Rapid analysis of very large scale parallel systems ODEs efficient implementation moments of: counts rewards passage and completion times error estimates large numbers of identical components described in GPEPA split-free splitting models Summary parallel solvers hybrid solutions impulse rewards time correlated moments optimisation nature of error new formalism complex synch. guards functional rates phase type Current work

178 49/60 Rapid analysis of very large scale parallel systems ODEs efficient implementation moments of: counts rewards passage and completion times error estimates large numbers of identical components described in GPEPA split-free splitting models Summary parallel solvers hybrid solutions impulse rewards time correlated moments optimisation nature of error new formalism complex synch. guards functional rates phase type Current work Applied to real systems

179 GPA: Download for free GPA tool [11] : [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. A new tool for the performance analysis of massively parallel computer systems. In: Eighth Workshop on Quantitative Aspects of Programming Languages (QAPL 2010), March 27-28, 2010, Paphos, Cyprus. Electronic Proceedings in Theoretical Computer Science. Mar url: 50/60

180 Fluid ODE generation using Population CTMCs 51/60

181 52/60 Populations CTMCs A Population continuous time Markov chain (PCTMC) consists of a finite set of components {1,..., N}, and a set T of transition classes.

182 52/60 Populations CTMCs A Population continuous time Markov chain (PCTMC) consists of a finite set of components {1,..., N}, and a set T of transition classes. Each state in a PCTMC is expressed as an integer vector X = (X 1,..., X N ) Z N

183 52/60 Populations CTMCs A Population continuous time Markov chain (PCTMC) consists of a finite set of components {1,..., N}, and a set T of transition classes. Each state in a PCTMC is expressed as an integer vector X = (X 1,..., X N ) Z N X i represents the current population level of a component i.

184 53/60 PCTMCs: Transition classes A transition class c = (r c, e c ) T describes a stochastic event Event c: X (t) X (t ) at rate r c

185 53/60 PCTMCs: Transition classes A transition class c = (r c, e c ) T describes a stochastic event Event c: X (t) X (t ) at rate r c 1. with exponentially distributed duration D at rate r c ( X (t)) where r c : Z N R is a rate function

186 53/60 PCTMCs: Transition classes A transition class c = (r c, e c ) T describes a stochastic event Event c: X (t) X (t ) at rate r c 1. with exponentially distributed duration D at rate r c ( X (t)) where r c : Z N R is a rate function 2. which changes the current population vector according to the change vector e c

187 53/60 PCTMCs: Transition classes A transition class c = (r c, e c ) T describes a stochastic event Event c: X (t) X (t ) at rate r c 1. with exponentially distributed duration D at rate r c ( X (t)) where r c : Z N R is a rate function 2. which changes the current population vector according to the change vector e c This gives us the following population dynamic formula: Event c: X (t + D) = X (t) + ec D exp(r c )

188 54/60 PCTMCs: Chemical reactions Similar to chemical reaction: s s k t t l at rate r( X )

189 54/60 PCTMCs: Chemical reactions Similar to chemical reaction: s s k t t l at rate r( X ) Change vector for this reaction would involve: e c = { 1,... 1, 1..., 1, 0,..., 0} }{{}}{{} k l

190 PCTMCs: Mean dynamics An important aspect of PCTMC models is that we can easily generate approximations to the evolution of the underlying stochastic process. [15] [15] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /60

191 PCTMCs: Mean dynamics An important aspect of PCTMC models is that we can easily generate approximations to the evolution of the underlying stochastic process. [15] In particular, the equation for a mean of population X i (t) is: d dt E[X i(t)] = (r j, e j ) T e ij r j ( X (t)) [15] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /60

192 56/60 ODE-based dynamics More generally PCTMCs permit the derivation of moments of the underlying stochastic process, i.e. moments of population levels d dt E[M( X (t))] = E[f M ( X (t))] where M( X ) defines the moment to be calculated.

193 56/60 ODE-based dynamics More generally PCTMCs permit the derivation of moments of the underlying stochastic process, i.e. moments of population levels d dt E[M( X (t))] = E[f M ( X (t))] where M( X ) defines the moment to be calculated. Mean of component 1: M( X ) = X 1 2nd moment of component 1: M( X ) = X 2 1 2nd joint moment of components 1 and 2: M( X ) = X 1 X 2

194 Higher moments The higher moment function is defined as: [16] f M ( X (t)) = c T(M( X (t) + e c )M( X (t))) r c ( X (t)) [16] Anton Stefanek. Efficient Computation of Performance Energy Trade-offs in Large Scale Markov Models. PhD thesis. Department of Computing, Imperial College London, /60

195 Higher moments The higher moment function is defined as: [16] f M ( X (t)) = c T(M( X (t) + e c )M( X (t))) r c ( X (t)) Key issue: achieving a closed set of equations with each quantity on right hand side of ODEs having a corresponding ODE. [16] Anton Stefanek. Efficient Computation of Performance Energy Trade-offs in Large Scale Markov Models. PhD thesis. Department of Computing, Imperial College London, /60

196 Higher moments The higher moment function is defined as: [16] f M ( X (t)) = c T(M( X (t) + e c )M( X (t))) r c ( X (t)) Key issue: achieving a closed set of equations with each quantity on right hand side of ODEs having a corresponding ODE. Leads to different dynamics: mean-field, mass action, min-closure, log-normal-closure [16] Anton Stefanek. Efficient Computation of Performance Energy Trade-offs in Large Scale Markov Models. PhD thesis. Department of Computing, Imperial College London, /60

197 58/60 Worked example: GPEPA Client = def (req, r req ).Client waiting Client waiting = def (data, r data ).Client think Client think = def (think, r think ).Client

198 58/60 Worked example: GPEPA Client = def (req, r req ).Client waiting Client waiting = def (data, r data ).Client think Client think = def (think, r think ).Client Server = def (req, r req ).Server get + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server

199 58/60 Worked example: GPEPA Client = def (req, r req ).Client waiting Client waiting = def (data, r data ).Client think Client think = def (think, r think ).Client Server = def (req, r req ).Server get + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

200 58/60 Worked example: GPEPA Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think Client think = def (think, r think ).Client Server = def (req, r req ).Server get + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

201 58/60 Worked example: GPEPA Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think C w (t) Client think = def (think, r think ).Client Server = def (req, r req ).Server get + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

202 58/60 Worked example: GPEPA Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think C w (t) Client think = def (think, r think ).Client C t (t) Server = def (req, r req ).Server get + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

203 58/60 Worked example: GPEPA Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think C w (t) Client think = def (think, r think ).Client C t (t) Server = def (req, r req ).Server get S(t) + (break, r break ).Server broken Server get = def (data, r data ).Server Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

204 58/60 Worked example: GPEPA Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think C w (t) Client think = def (think, r think ).Client C t (t) Server = def (req, r req ).Server get S(t) + (break, r break ).Server broken Server get = def (data, r data ).Server S g (t) Server broken = def (reset, r reset ).Server CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

205 58/60 Worked example: GPEPA Client = def (req, r req ).Client waiting C(t) Client waiting = def (data, r data ).Client think C w (t) Client think = def (think, r think ).Client C t (t) Server = def (req, r req ).Server get S(t) + (break, r break ).Server broken Server get = def (data, r data ).Server S g (t) Server broken = def (reset, r reset ).Server S b (t) CS(c, s) = Clients{Client[c]} {req,data} Servers{Server[s]}

206 59/60 Worked example: PCTMC In total, there are 5 transition classes: req : data : think : break : reset :

207 59/60 Worked example: PCTMC In total, there are 5 transition classes: req : C(t) + S(t) C w (t) + S g (t) data : think : break : reset : at r req min(c(t), S(t))

208 59/60 Worked example: PCTMC In total, there are 5 transition classes: req : C(t) + S(t) C w (t) + S g (t) data : C w (t) + S g (t) C t (t) + S(t) think : break : reset : at r req min(c(t), S(t)) at r data min(c w (t), S g (t))

209 59/60 Worked example: PCTMC In total, there are 5 transition classes: req : C(t) + S(t) C w (t) + S g (t) data : C w (t) + S g (t) C t (t) + S(t) think : C t (t) C(t) at r think C t (t) break : reset : at r req min(c(t), S(t)) at r data min(c w (t), S g (t))

210 59/60 Worked example: PCTMC In total, there are 5 transition classes: req : C(t) + S(t) C w (t) + S g (t) data : C w (t) + S g (t) C t (t) + S(t) think : C t (t) C(t) at r think C t (t) break : S(t) S b (t) at r break S(t) reset : at r req min(c(t), S(t)) at r data min(c w (t), S g (t))

211 59/60 Worked example: PCTMC In total, there are 5 transition classes: req : C(t) + S(t) C w (t) + S g (t) data : C w (t) + S g (t) C t (t) + S(t) think : C t (t) C(t) at r think C t (t) break : S(t) S b (t) at r break S(t) reset : S b (t) S(t) at r reset S b (t) at r req min(c(t), S(t)) at r data min(c w (t), S g (t))

212 59/60 Worked example: PCTMC In total, there are 5 transition classes: req : C(t) + S(t) C w (t) + S g (t) data : C w (t) + S g (t) C t (t) + S(t) think : C t (t) C(t) at r think C t (t) break : S(t) S b (t) at r break S(t) reset : S b (t) S(t) at r reset S b (t) at r req min(c(t), S(t)) at r data min(c w (t), S g (t)) Then apply PCTMC ODE generation rules to get a fluid GPEPA model.

213 Even more exciting fluid analysis 60/60

214 Scalable passage-time analysis 12/30

215 Scalable passage-time analysis Passage-time distributions are key for specifying service level agreements (SLAs), e.g.: [7,8] file should be transferred within 2 seconds, 95% of the time [7] Richard A. Hayden, Anton Stefanek, and Jeremy T. Bradley. Fluid computation of passage time distributions in large Markov models. In: Theoretical Computer Science (2012), pp doi: /j.tcs [8] Richard A. Hayden, Jeremy T. Bradley, and Allan Clark. Performance Specification and Evaluation with Unified Stochastic Probes and Fluid Analysis. In: IEEE Transactions on Software Engineering 99.PrePrints (2012). doi: /TSE /30

216 13/30 Scalable passage-time analysis Passage-time distributions are key for specifying service level agreements (SLAs), e.g.: connection should be established within 0.25 seconds, 99% of the time

217 13/30 Scalable passage-time analysis Passage-time distributions are key for specifying service level agreements (SLAs), e.g.: connection should be established within 0.25 seconds, 99% of the time We consider two classes of passage-time query:

218 13/30 Scalable passage-time analysis Passage-time distributions are key for specifying service level agreements (SLAs), e.g.: connection should be established within 0.25 seconds, 99% of the time We consider two classes of passage-time query: Individual passage times: track the time taken for an individual to complete a task

219 13/30 Scalable passage-time analysis Passage-time distributions are key for specifying service level agreements (SLAs), e.g.: connection should be established within 0.25 seconds, 99% of the time We consider two classes of passage-time query: Individual passage times: track the time taken for an individual to complete a task Global passage times: track the time taken for all of a large number of individuals to complete a task

220 13/30 Scalable passage-time analysis Passage-time distributions are key for specifying service level agreements (SLAs), e.g.: connection should be established within 0.25 seconds, 99% of the time We consider two classes of passage-time query: Individual passage times: track the time taken for an individual to complete a task Direct approximation to the entire CDF Global passage times: track the time taken for all of a large number of individuals to complete a task

221 13/30 Scalable passage-time analysis Passage-time distributions are key for specifying service level agreements (SLAs), e.g.: connection should be established within 0.25 seconds, 99% of the time We consider two classes of passage-time query: Individual passage times: track the time taken for an individual to complete a task Direct approximation to the entire CDF Global passage times: track the time taken for all of a large number of individuals to complete a task Moment-derived bounds on CDF

222 Individual passage times [7] Client Server fail proc req Client wait reset Server fail res res req Client proc Server proc N C N S How long does it take a single client to make a request, receive a response and process it? [7] Richard A. Hayden, Anton Stefanek, and Jeremy T. Bradley. Fluid computation of passage time distributions in large Markov models. In: Theoretical Computer Science (2012), pp doi: /j.tcs /30

223 14/30 Individual passage times Client Client Server fail proc req Client wait proc req Client wait reset Server fail res res res req Client proc 1 Client proc N C 1 Server proc N S How long does it take a single client to make a request, receive a response and process it?

224 14/30 Individual passage times proc C(t) Client req Client wait Client proc req Client wait proc Client req Client wait Server fail reset Server fail res res res res req Client proc Client proc 1 Client proc N C 1 Server proc N S How long does it take a single client to make a request, receive a response and process it?

225 14/30 Individual passage times proc C(t) Client req Client wait Client proc req Client wait proc Client req Client wait Server fail reset Server fail res res res res req Client proc Client proc Client proc Server proc 1 N C 1 T := inf{t 0 : C(t) = Client }, given that C(0) = Client N S

226 14/30 Individual passage times proc C(t) Client req Client wait Client proc req Client wait proc Client req Client wait Server fail reset Server fail res res res res req Client proc Client proc Client proc Server proc 1 N C 1 T := inf{t 0 : C(t) = Client }, given that C(0) = Client N S P{T t} = P{C(t) {Client, Client wait, Client proc}}

227 14/30 Individual passage times proc C(t) Client req Client wait Client proc req Client wait proc Client req Client wait Server fail reset Server fail res res res res req Client proc Client proc Client proc Server proc 1 N C 1 T := inf{t 0 : C(t) = Client }, given that C(0) = Client N S P{T t} = P{C(t) {Client, Client wait, Client proc}} = E[1 {C(t)=Client }] + E[1 {C(t)=Client wait }] + E[1 {C(t)=Client proc }]

228 14/30 Individual passage times proc C(t) Client req Client wait Client proc req Client wait proc Client req Client wait Server fail reset Server fail res res res res req Client proc Client proc Client proc Server proc 1 N C 1 T := inf{t 0 : C(t) = Client }, given that C(0) = Client N S P{T t} = P{C(t) {Client, Client wait, Client proc}} = E[1 {C(t)=Client }] + E[1 {C(t)=Client wait }] + E[1 {C(t)=Client proc }] = E[N Client (t)] + E[N Client wait (t)] + E[N Client proc (t)]

229 14/30 Individual passage times proc C(t) Client req Client wait Client proc req Client wait proc Client req Client wait Server fail reset Server fail res res res res req Client proc Client proc Client proc Server proc 1 N C 1 T := inf{t 0 : C(t) = Client }, given that C(0) = Client N S P{T t} = P{C(t) {Client, Client wait, Client proc}} = E[1 {C(t)=Client }] + E[1 {C(t)=Client wait }] + E[1 {C(t)=Client proc }] = E[N Client (t)] + E[N Client wait (t)] + E[N Client proc (t)] P{T t} v Client (t) + v Client wait (t) + v Client proc (t)

230 Example individual passage time Probability N C = 10, N S = 6 N C = 20, N S = 12 N C = 50, N S = 30 N C = 100, N S = 60 N C = 200, N S = 120 ODE approximation Time, t 15/30

231 Global passage times [7,9] Client Server fail proc req Client wait reset Server fail res res req Client proc Server proc N C N S How long does it take for half of the clients to make a request, receive a response and process it? [7] Richard A. Hayden, Anton Stefanek, and Jeremy T. Bradley. Fluid computation of passage time distributions in large Markov models. In: Theoretical Computer Science (2012), pp doi: /j.tcs [9] Rena Bakhshi et al. Mean-Field Analysis for the Evaluation of Gossip Protocols. In: QEST 09, Proceedings of the 5th IEEE Conference on the Quantitative Evaluation of Systems. IEEE Computer Society, 2009, pp /30

232 16/30 Global passage times proc Client req Client wait proc Client req Client wait Server fail reset Server fail res res res req Client proc Client proc NC Server proc N S How long does it take for half of the clients to make a request, receive a response and process it?

233 16/30 Global passage times proc Client req Client wait proc Client req Client wait Server fail reset Server fail res res res req Client proc Client proc NC Server proc N S T := inf{t 0 : N C (t) + N C w (t) + N C p (t) N C /2}

234 16/30 Global passage times proc Client req Client wait proc Client req Client wait Server fail reset Server fail res res res req Client proc Client proc NC Server proc N S T := inf{t 0 : N C (t) + N C w (t) + N C p (t) N C /2} Point-mass approximation: T inf{t 0 : v C (t) + v C w (t) + v C p (t) N C /2}

235 16/30 Global passage times Probability N C = 10, N S = 6 N C = 20, N S = 12 N C = 50, N S = 30 N C = 300, N S = 180 N C = 500, N S = Time, t

236 16/30 Global passage times proc Client req Client wait proc Client req Client wait Server fail reset Server fail res res res req Client proc Client proc Server proc NC N S Point-mass approximation: T inf{t 0 : v C (t) + v C w (t) + v C p (t) N C /2} Approximation is very coarse Cannot be applied directly to the same question for all clients

237 Global passage times moment bounds proc Client req Client wait proc Client req Client wait Server fail reset Server fail res res res req Client proc Client proc NC Server proc N S Moment approximations to component counts contain information about the distribution of T [7] [7] Richard A. Hayden, Anton Stefanek, and Jeremy T. Bradley. Fluid computation of passage time distributions in large Markov models. In: Theoretical Computer Science (2012), pp doi: /j.tcs /30

238 Global passage times moment bounds proc Client req Client wait proc Client req Client wait Server fail reset Server fail res res res req Client proc Client proc NC Server proc N S Moment approximations to component counts contain information about the distribution of T Reduced moment problem find maximum and minimum bounding distributions subject to limited moment information [10] [10] Arpád Tari, Miklós Telek, and Peter Buchholz. A Unified Approach to the Moments-based Distribution Estimation Unbounded Support. In: EPEW 05, European Performance Engineering Workshop. Vol Lecture Notes in Computer Science. Versailles: Springer, 2005, pp /30

239 Global passage bounds first moments [7] Half of the clients: Three quarters of the clients: Probability CDF 0.2 Upper Lower Time, t 1 Probability All of the clients: CDF Upper 0.2 Lower Time, t 0.8 Probability CDF 0.2 Upper Lower Time, t [7] Richard A. Hayden, Anton Stefanek, and Jeremy T. Bradley. Fluid computation of passage time distributions in large Markov models. In: Theoretical Computer Science (2012), pp doi: /j.tcs /30

240 Global passage bounds higher moments [7] Probability Time, t 1st order 2nd order 4th order CDF [7] Richard A. Hayden, Anton Stefanek, and Jeremy T. Bradley. Fluid computation of passage time distributions in large Markov models. In: Theoretical Computer Science (2012), pp doi: /j.tcs /30

241 Scalable analysis of accumulated reward measures 20/30

242 Accumulated reward measures [11] Cost, energy, heat,... Constant rate [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

243 Accumulated reward measures [11] Cost, energy, heat,... Constant rate Server e t Server proc e t [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

244 Accumulated reward measures [11] Cost, energy, heat,... Constant rate Server e t Server proc e t state at t Server proc r Sp Server r S Server fail Time, t [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

245 Accumulated reward measures [11] Cost, energy, heat,... Constant rate Server e t Server proc e t state at t Server proc r Sp Server r S Server fail Time, t [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

246 Accumulated reward measures [11] Cost, energy, heat,... Constant rate Server e t Server proc e t state at t N S (t) Server proc r Sp 20 Count 10 Server r S Server fail Time, t Time, t [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

247 Accumulated reward measures [11] Cost, energy, heat,... Constant rate Server e t Server proc e t state at t N S (t) Server proc r Sp 20 r S Count 10 Server r S Server fail Time, t Time, t [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

248 Accumulated reward measures [11] Cost, energy, heat,... Constant rate Server e t Server proc e t state at t N S (t) Server proc r Sp 20 r S N Sp (t) Count 10 Server r S Server fail Time, t Time, t [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

249 Accumulated reward measures [11] Cost, energy, heat,... Constant rate Server e t Server proc e t state at t N S (t) Server proc r Sp 20 r S r Sp N Sp (t) Count 10 Server r S Server fail Time, t Time, t [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

250 Accumulated reward measures [11] Cost, energy, heat,... Constant rate Server e t Server proc e t state at t N S (t) Server proc r Sp 20 r S r Sp N Sp (t) Count 10 Server r S Server fail Time, t Time, t t t total energy(t) = r S N S (u) du + r Sp N Sp (u) du 0 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

251 Moment approximations of accumulated rewards [11] 150 Accumulated quantity t 0 Sp(u) du Time, t Simulation also very costly for rewards [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

252 Moment approximations of accumulated rewards [11] 150 Accumulated quantity t 0 Sp(u) du Time, t Simulation also very costly for rewards [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

253 Moment approximations of accumulated rewards [11] 150 Accumulated quantity t 0 Sp(u) du Time, t Simulation also very costly for rewards [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

254 Moment approximations of accumulated rewards [11] 150 Accumulated quantity Time, t [ ] E t 0 Sp(u) du Simulation also very costly for rewards Can extend the ODE system for count moments with ODEs for moments of accumulated counts: [ d t ] dt E N Sp (u) du = 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

255 Moment approximations of accumulated rewards [11] 150 Accumulated quantity Time, t [ ] E t 0 Sp(u) du Simulation also very costly for rewards Can extend the ODE system for count moments with ODEs for moments of accumulated counts: [ d t t ] dt E N Sp (u) du N S (u) du = 0 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

256 Moment approximations of accumulated rewards [11] 150 Accumulated quantity Time, t [ ] E t 0 Sp(u) du Simulation also very costly for rewards Can extend the ODE system for count moments with ODEs for moments of accumulated counts: [ d t t ] dt E N Sp (u) du N S (u) du = 0 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

257 Moment approximations of accumulated rewards [11] 150 Accumulated quantity Time, t [ ] E t 0 Sp(u) du Simulation also very costly for rewards Can extend the ODE system for count moments with ODEs for moments of accumulated counts: [ d t t ] dt E N Sp (u) du N S (u) du = 0 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

258 Moment approximations of accumulated rewards [11] 150 Accumulated quantity Time, t [ ] E t 0 Sp(u) du Simulation also very costly for rewards Can extend the ODE system for count moments with ODEs for moments of accumulated counts: [ d t t ] dt E N Sp (u) du N S (u) du = 0 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

259 Moment approximations of accumulated rewards [11] First-order moments [ d t ] dt E N S (u) du = 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

260 Moment approximations of accumulated rewards [11] First-order moments [ d t ] dt E N S (u) du = E[N S (t)] 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

261 Moment approximations of accumulated rewards [11] First-order moments [ d t ] dt E N S (u) du = E[N S (t)] 0 First-order moments d dt E[N S(t)] = [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

262 Moment approximations of accumulated rewards [11] First-order moments [ d t ] dt E N S (u) du = E[N S (t)] 0 First-order moments d dt E[N S(t)] = Second-order moments [ ( d t ) ] 2 dt E N S (u) du = 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

263 Moment approximations of accumulated rewards [11] First-order moments [ d t ] dt E N S (u) du = E[N S (t)] 0 First-order moments d dt E[N S(t)] = Second-order moments [ ( d t ) ] 2 dt E N S (u) du 0 [ t ] = 2E N S (t) N S (u) du 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

264 Moment approximations of accumulated rewards [11] First-order moments [ d t ] dt E N S (u) du = E[N S (t)] 0 First-order moments d dt E[N S(t)] = Second-order moments [ ( d t ) ] 2 dt E N S (u) du 0 Combined moments [ d t ] dt E N S (t) N S (u) du = 0 [ t ] = 2E N S (t) N S (u) du 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

265 Moment approximations of accumulated rewards [11] First-order moments [ d t ] dt E N S (u) du = E[N S (t)] 0 First-order moments d dt E[N S(t)] = Second-order moments [ ( d t ) ] 2 dt E N S (u) du 0 Combined moments [ d t ] dt E N S (t) N S (u) du 0 [ t ] = 2E N S (t) N S (u) du 0 [ t ] = E N S (t) N S (u) du + + E[NS 2 (t)] 0 [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

266 Moment approximations of accumulated rewards [11] First-order moments [ d t ] dt E N S (u) du = E[N S (t)] 0 Second-order moments [ ( d t ) ] 2 dt E N S (u) du 0 Combined moments [ d t ] dt E N S (t) N S (u) du 0 [ t = 2E N S (t) N S (u) du 0 First-order moments d dt E[N S(t)] = Second-order moments d dt E[N S(t)N Sp (t)] = [ t ] = E N S (t) N S (u) du + + E[NS 2 (t)] 0 ] [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

267 Moment approximations of accumulated rewards [11] First-order moments [ d t ] dt E N S (u) du = E[N S (t)] 0 Second-order moments [ ( d t ) ] 2 dt E N S (u) du 0 Combined moments [ d t ] dt E N S (t) N S (u) du 0 [ t = 2E N S (t) N S (u) du 0 First-order moments d dt E[N S(t)] = Second-order moments d dt E[N S(t)N Sp (t)] = [ t ] = E N S (t) N S (u) du + + E[NS 2 (t)] 0 ] [11] Anton Stefanek, Richard A. Hayden, and Jeremy T. Bradley. Fluid analysis of energy consumption using rewards in massively parallel Markov models. In: 2nd ACM/SPEC International Conference on Performance Engineering (ICPE). 2011, pp doi: / /30

268 24/30 Trade-off between energy and performance Client Server fail proc req Client wait reset Server fail res req Client proc res Server proc N C N S

269 24/30 Trade-off between energy and performance Client Server fail proc req Client wait reset Server fail res req Client proc res Server proc N C N S 1 Client serviced before t 150 E[total energy(t)] Probability Energy Time, t Time, t

270 24/30 Trade-off between energy and performance Client Server fail proc req Client wait reset Server fail res req Client proc res Server proc N C N S 1 Client serviced before t 150 E[total energy(t)] Probability SLA 7s 0.99 Energy Time, t Time, t

271 24/30 Trade-off between energy and performance Client Server fail proc req Client wait res Client proc res reset fail Server Server sleep sleep/wakeup req Server proc N C N S 1 Client serviced before t 150 E[total energy(t)] Probability SLA 7s 0.99 Energy Time, t Time, t

272 24/30 Trade-off between energy and performance Client Server fail proc req Client wait res Client proc res reset fail Server Server sleep sleep/wakeup req Server proc N C N S 1 Client serviced before t 150 E[total energy(t)] Probability SLA 7s 0.99 Energy Time, t Time, t

273 25/30 Trade-off between energy and performance Scalable analysis allows exploration of many configurations (N S, sleep rate)

274 25/30 Trade-off between energy and performance Scalable analysis allows exploration of many configurations (N S, sleep rate) Minimise energy consumption while satisfying SLAs

275 25/30 Trade-off between energy and performance Scalable analysis allows exploration of many configurations (N S, sleep rate) Minimise energy consumption while satisfying SLAs Energy consumption 1,200 1, N S r sleep Individual passage-time SLA:

276 25/30 Trade-off between energy and performance Scalable analysis allows exploration of many configurations (N S, sleep rate) Minimise energy consumption while satisfying SLAs Energy consumption 1,200 1, SLA met N S r sleep Individual passage-time SLA: clients must finish in at most 7s 99% of the time

277 25/30 Trade-off between energy and performance Scalable analysis allows exploration of many configurations (N S, sleep rate) Minimise energy consumption while satisfying SLAs Energy consumption 1,200 1, SLA met N S r sleep Individual passage-time SLA: clients must finish in at most 7s 99% of the time

278 25/30 Trade-off between energy and performance Scalable analysis allows exploration of many configurations (N S, sleep rate) Minimise energy consumption while satisfying SLAs Energy consumption 1,200 1, SLA met N S r sleep Individual passage-time SLA: clients must finish in at most 7s 99% of the time

279 25/30 Trade-off between energy and performance Scalable analysis allows exploration of many configurations (N S, sleep rate) Minimise energy consumption while satisfying SLAs Energy consumption 1,200 1, SLA met N S r sleep Individual passage-time SLA: clients must finish in at most 7s 99.5% of the time

280 25/30 Trade-off between energy and performance Scalable analysis allows exploration of many configurations (N S, sleep rate) Minimise energy consumption while satisfying SLAs Energy consumption 1,200 1, SLA met N S r sleep Individual passage-time SLA: clients must finish in at most 7s 99.5% of the time

281 Non-Markovian models 26/30

282 27/30 Non-Markovian models Distributions more general than exponential are required to construct realistic models, for example: Deterministic timeouts in protocols or hardware Heavy-tailed service-time distributions

283 27/30 Non-Markovian models Distributions more general than exponential are required to construct realistic models, for example: Deterministic timeouts in protocols or hardware Heavy-tailed service-time distributions Phase-type approximation is one approach, but can lead to significant increase in a component s local state-space size A 100-phase Erlang approximation to a deterministic distribution of duration 1 has a probability of about 32% of lying outside of [0.9, 1.1]

284 27/30 Non-Markovian models Distributions more general than exponential are required to construct realistic models, for example: Deterministic timeouts in protocols or hardware Heavy-tailed service-time distributions Phase-type approximation is one approach, but can lead to significant increase in a component s local state-space size A 100-phase Erlang approximation to a deterministic distribution of duration 1 has a probability of about 32% of lying outside of [0.9, 1.1] In the case of deterministic distributions, mean-field approach can be generalised using delay differential equations

285 Software update model with deterministic timeouts [12] d γ ρ c e λ ρ βn a(t) N λ a b ρ Old node Updated node [12] Richard A. Hayden. Mean-field approximations for performance models with generally-timed transitions. In: ACM SIGMETRICS Performance Evaluation Review 39.3 (2011), pp doi: / /30

286 Software update model with deterministic timeouts [12] d γ ρ c e λ ρ βn a(t) N λ a b ρ Old node Updated node Ė[N c (t)] = ρe[n c (t)] β N E[N c(t)n a (t)] + λe[n e (t)] [ ( t ) ] βn a (s) E 1 {t γ} λn e (t γ) exp ds exp( ργ) }{{} t γ N Rate of determ. clocks starting at t γ [12] Richard A. Hayden. Mean-field approximations for performance models with generally-timed transitions. In: ACM SIGMETRICS Performance Evaluation Review 39.3 (2011), pp doi: / /30

287 Software update model with deterministic timeouts [12] d γ ρ c e λ ρ βn a(t) N λ a b ρ Old node Updated node Ė[N c (t)] = ρe[n c (t)] β N E[N c(t)n a (t)] + λe[n e (t)] [ ( t βn a (s) E 1 {t γ} λn e (t γ) exp N t γ ) ds exp( ργ) } {{ } Prob. that timeout occurs before node updated or went off ] [12] Richard A. Hayden. Mean-field approximations for performance models with generally-timed transitions. In: ACM SIGMETRICS Performance Evaluation Review 39.3 (2011), pp doi: / /30

288 Software update model with deterministic timeouts [12] d γ ρ c e λ ρ βn a(t) N λ a b ρ Old node Updated node Ė[N c (t)] ρe[n c (t)] β N E[N c(t)]e[n a (t)] + λe[n e (t)] ( t 1 {t γ} λe[n e (t γ)] exp t γ βe[n a (s)] N ) ds exp( ργ) [12] Richard A. Hayden. Mean-field approximations for performance models with generally-timed transitions. In: ACM SIGMETRICS Performance Evaluation Review 39.3 (2011), pp doi: / /30

289 Software update model with deterministic timeouts [12] d γ ρ c e λ ρ βn a(t) N λ a b ρ Old node Updated node v c (t) = ρv c (t) β N v c(t)v a (t) + λv e (t) ( t βv a (s) 1 t γ λv e (t γ) exp N t γ ) ds exp( ργ) [12] Richard A. Hayden. Mean-field approximations for performance models with generally-timed transitions. In: ACM SIGMETRICS Performance Evaluation Review 39.3 (2011), pp doi: / /30

290 29/30 Software update model with deterministic timeouts Rescaled component count Nodes in state c Nodes in state d Nodes in state e Time, t Rescaled component count Nodes in state a Nodes in state b Time, t

291 Summary Fluid analysis provides a scalable analysis framework for massively-parallel performance models, that is able to capture: Arbitrary moments of component counts Passage-time measures Accumulated reward measures Certain forms of non-markovian timing with implementation in the freely-available GPA tool /30

292 Thank you! 2 2 Many thanks to Richard Hayden and Anton Stefanek for their expertise with pgf and pgfplots and their help with this presentation. They also did a substantial portion of the research! 31/30